Add amplitude damping noise model (#497)

[NandanPatel24]

Summary

• Implements amplitude damping noise (issue Implement amplitude damping noise model #497), modeling energy relaxation (T₁ decay) where population flows from the excited state |1⟩ to the ground state |0⟩ with probability γ. Adds the single- and two-qubit channels, the Noise element classes, and an AmplitudeDampingNoiseModel, following DepolarisingNoiseModel as the reference.

Closes #497.

Changes

• graphix/channels.py — adds amplitude_damping_channel(prob) and two_qubit_amplitude_damping_channel(prob), returning KrausChannel objects. Single-qubit operators are K₁ = diag(1, √(1−γ)) and K₂ = [[0, √γ], [0, 0]]; the two-qubit version is the tensor product of two independent single-qubit channels (four operators {Kᵢ⊗Kⱼ}). Unlike the existing Pauli-based channels these operators aren't scalar multiples of Pauli matrices, so they're built explicitly with coef=1.0.

• graphix/noise_models/amplitude_damping.py (new) — defines AmplitudeDampingNoise, TwoQubitAmplitudeDampingNoise (both deriving from Noise), and AmplitudeDampingNoiseModel (deriving from NoiseModel). The model mirrors DepolarisingNoiseModel's command-injection logic. confuse_result returns the result unchanged, since amplitude damping is a purely quantum channel with no classical readout-error component; readout error can be added by composing with another model via ComposeNoiseModel. (This is why the model omits the measure_error_prob parameter that the depolarising model carries.)

• graphix/noise_models/init.py — registers the three new classes for top-level import, alongside the depolarising exports.

A convention note

• The backend (density_matrix.py) applies channels as ρ′ = Σᵢ KᵢρKᵢ† (verified in evolve/evolve_single). The KrausChannel docstring states the daggered-left form Σᵢ Kᵢ†ρKᵢ — for the existing Hermitian Pauli channels this makes no difference, but amplitude damping's K₂ is non-Hermitian, so the operators here are written for the backend's actual KρK† convention (downward |1⟩ → |0⟩ decay). Happy to also fix the docstring in this PR if maintainers prefer.

Tests and soundness

The suite is split between proving the channel matches the Kraus formula (random-state and by-hand-sum tests) and proving it is amplitude damping specifically (deterministic basis-state tests exploiting the channel's asymmetry — these are the soundness argument the issue asks for).

tests/test_kraus.py

• test_amplitude_damping_channel — single-qubit channel builds the correct K₁, K₂ operators.
• test_2_qubit_amplitude_damping_channel — two-qubit channel builds the correct four tensor-product operators.

tests/test_density_matrix.py(https://github.com/TeamGraphix/graphix/blob/master/tests/test_density_matrix.py)

• test_apply_amplitude_damping_channel — channel application matches the by-hand Σ KᵢρKᵢ† sum, on a single qubit and embedded in a larger random register.
• test_amplitude_damping_ground_state_fixed — |0⟩⟨0| is unchanged for any γ (ground state can't decay).
• test_amplitude_damping_excited_state_decays — |1⟩⟨1| → (1−γ)|1⟩⟨1| + γ|0⟩⟨0|, the directional T₁ signature.
• test_amplitude_damping_coherence_decay — off-diagonal coherences scale by √(1−γ), distinguishing amplitude damping from dephasing.
• test_apply_two_qubit_amplitude_damping_channel — the two-qubit channel equals independent damping on each factor.

tests/test_noise_model.py((https://github.com/TeamGraphix/graphix/blob/master/tests/test_noise_model.py)

• test_amplitude_damping_command_injection — noise is injected at the correct positions (after N, after E with two-qubit noise, before M, domain-conditioned on X/Z).
• test_amplitude_damping_confuse_result_is_identity — the model introduces no classical readout error.
• test_compose_amplitude_damping_depolarising_transpile — composing with a depolarising model injects both noises with the correct type and probability at every command position.
• test_compose_amplitude_damping_depolarising_simulation — a composed model with both at default (zero) parameters reproduces the ideal statevector.

All checks pass locally (ruff, ruff-format, mypy, pyright, pytest).

Per unitaryHACK's AI-use guidelines: AI assistance was used while developing this contribution, especially to format this PR content, and for debugging. All ideas remain my own